Problem: Let $\alpha$ and $\beta$ be complex numbers such that $\alpha + \beta$ and $i(\alpha - 2 \beta)$ are both positive real numbers.  If $\beta = 3 + 2i,$ compute $\alpha.$
Answer: Let $x = \alpha + \beta$ and $y = i (\alpha - 2 \beta).$  Then $\alpha - 2 \beta = \frac{y}{i} = -yi.$  Solving for $\alpha$ and $\beta,$ we get
\begin{align*}
\alpha &= \frac{2}{3} x - \frac{y}{3} i, \\
\beta &= \frac{1}{3} x + \frac{y}{3} i.
\end{align*}Since $x$ and $y$ are real, and $\beta = 3 + 2i,$ $x = 9$ and $y = 6.$  Then $\alpha = \boxed{6 - 2i}.$